Nonlinear stochastic differential equations: A renormalization group approach to direct calculation of moments

While linear stochastic differential equations are exactly solvable, the solutions for nonlinear equations are traditionally sought from the corresponding Fokker-Planck description. Based on the separation of deterministic and stochastic time scales in the dynamics, a method for direct calculation of the mean and variance of the distribution for nonlinear stochastic differential equations is proposed using the renormalization group (RG) technique. We have shown how nonlinearity and its interplay with noise brings corrections to the frequency of the dynamical system, as reflected in the RG flow equations for amplitude and phase. Two broad classes of nonlinear systems were explored, one with linear dissipation, nonlinear potential, and internal noise obeying fluctuation-dissipation relation, and the other with nonlinear dissipation and linear potential, allowing a limit cycle solution subjected to external noise. We analyzed the mean-square displacement as a measure of diffusive behavior and determined the stability threshold of the limit cycle against the external noise. Our theory is compared with full numerical simulations.